The magnetic flux, represented by the symbol Φ, threading some contour or loop is defined as the magnetic field B multiplied by the loop area S, i.e. Φ = B ⋅ S. Both B and S can be arbitrary, meaning Φ can be as well. However, if one deals with the superconducting loop or a hole in a bulk superconductor, the magnetic flux threading such a hole/loop is actually quantized. The (superconducting) magnetic flux quantum Φ0 = h/(2e) ≈ 2.067833848...×10−15 Wb is a combination of fundamental physical constants: the Planck constant h and the electron charge e. Its value is, therefore, the same for any superconductor. The phenomenon of flux quantization was discovered experimentally by B. S. Deaver and W. M. Fairbank and, independently, by R. Doll and M. Näbauer, in 1961. The quantization of magnetic flux is closely related to the Little–Parks effect, but was predicted earlier by Fritz London in 1948 using a phenomenological model.
The inverse of the flux quantum, 1/Φ0, is called the Josephson constant, and is denoted KJ. It is the constant of proportionality of the Josephson effect, relating the potential difference across a Josephson junction to the frequency of the irradiation. The Josephson effect is very widely used to provide a standard for high-precision measurements of potential difference, which (from 1990 to 2019) were related to a fixed, conventional value of the Josephson constant, denoted KJ-90. With the 2019 redefinition of SI base units, the Josephson constant has an exact value of KJ = 483597.84841698... GHz⋅V−1, which replaces the conventional value KJ-90.
The following physical equations use SI units. In CGS units, a factor of c would appear.
The superconducting properties in each point of the superconductor are described by the complex quantum mechanical wave function Ψ(r,t) — the superconducting order parameter. As any complex function Ψ can be written as Ψ = Ψ0eiθ, where Ψ0 is the amplitude and θ is the phase. Changing the phase θ by 2πn will not change Ψ and, correspondingly, will not change any physical properties. However, in the superconductor of non-trivial topology, e.g. superconductor with the hole or superconducting loop/cylinder, the phase θ may continuously change from some value θ0 to the value θ0 + 2πn as one goes around the hole/loop and comes to the same starting point. If this is so, then one has n magnetic flux quanta trapped in the hole/loop, as shown below:
Here, the wave function is the Ginzburg–Landau order parameter:
Plugging into the expression of probability current, one obtains:
While inside the body of the superconductor, the current density J is zero; Therefore:
Integrating around the hole/loop using Stokes' theorem and gives:
Now, because the order parameter must return to the same value when the integral goes back to the same point, we have:
Due to the Meissner effect, the magnetic induction B inside the superconductor is zero. More exactly, magnetic field H penetrates into a superconductor over a small distance called London's magnetic field penetration depth (denoted λL and usually ≈ 100 nm). The screening currents also flow in this λL-layer near the surface, creating magnetization M inside the superconductor, which perfectly compensates the applied field H, thus resulting in B = 0 inside the superconductor.
The magnetic flux frozen in a loop/hole (plus its λL-layer) will always be quantized. However, the value of the flux quantum is equal to Φ0 only when the path/trajectory around the hole described above can be chosen so that it lays in the superconducting region without screening currents, i.e. several λL away from the surface. There are geometries where this condition cannot be satisfied, e.g. a loop made of very thin (≤ λL) superconducting wire or the cylinder with the similar wall thickness. In the latter case, the flux has a quantum different from Φ0.
Flux quantization also plays an important role in the physics of type II superconductors. When such a superconductor (now without any holes) is placed in a magnetic field with the strength between the first critical field Hc1 and the second critical field Hc2, the field partially penetrates into the superconductor in a form of Abrikosov vortices. The Abrikosov vortex consists of a normal core—a cylinder of the normal (non-superconducting) phase with a diameter on the order of the ξ, the superconducting coherence length. The normal core plays a role of a hole in the superconducting phase. The magnetic field lines pass along this normal core through the whole sample. The screening currents circulate in the λL-vicinity of the core and screen the rest of the superconductor from the magnetic field in the core. In total, each such Abrikosov vortex carries one quantum of magnetic flux Φ0.
Measuring the magnetic fluxEdit
Prior to the 2019 redefinition of the SI base units, the magnetic flux quantum was measured with great precision by exploiting the Josephson effect. When coupled with the measurement of the von Klitzing constant RK = h/e2, this provided the most accurate values of Planck's constant h obtained until 2019. This may be counterintuitive, since h is generally associated with the behavior of microscopically small systems, whereas the quantization of magnetic flux in a superconductor and the quantum Hall effect are both emergent phenomena associated with thermodynamically large numbers of particles.
As a result of the 2019 redefinition of the SI base units, the Planck constant h has a fixed value 6.62607015×10−34 J⋅Hz−1, which, together with the definitions of the second and the metre, provides the official definition of the kilogram. Furthermore, the elementary charge also has a fixed value of e = 1.602176634×10−19 C to define the ampere. Therefore, both the Josephson constant KJ=(2e)/h and the von Klitzing constant RK = h/e2 have fixed values, and the Josephson effect along with the von Klitzing quantum Hall effect becomes the primary mise en pratique for the definition of the ampere and other electric units in the SI.
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